Oscillating Dumbbell Dynamics in Viscous Fluids: Analytical Insights

The aim of this chapter is to investigate analytically the motion of oscillating dumbbell, two micro-spheres connected by a spring, in a viscous incompressible fluid at low Reynolds number. The analytical approach used in the study offers a detailed and precise understanding of the motion characteristics, which can help improve the design and optimization of systems involving oscillatory motion in viscous environments. The oscillating dumbbell consists of one conducting sphere and assumed to be actively in motion under the action of an external oscillator field while the other is non-conducting sphere. As result, the oscillating dumbbell moves due to the induced ow oscillation of the surrounding fluid. The fluid ow past the spheres is described by the Stokes equation and the governing equation in the vector form for the oscillating dumbbell is solved asymptotically using the two-timing method. For illustrations, applying a simple oscillatory external field, a systematic description of the average velocity of the oscillating dumbbell is formulated. The trajectory of the oscillating dumbbell was found to be inversely proportional to the frequency of the external field, and the results demonstrated that the oscillating dumbbell moves in a circular path with a speed that decreases inversely with the length of the spring. It is worth noting that the average velocities of the oscillating dumbbell is given in the most general form which could be applicable to study a full three-dimensional problem.